Cohomology, Function Fields and Anabelian Geometry

代数学和算术几何研究多项式和/或丢番图方程的解，而伽罗瓦理论研究此类方程的对称性。通过考虑这些算术/几何对象到双有理等价，人们通常可以获得关于它们之间关系的新信息。事实上，算术/代数几何中的许多重要问题都具有双有理性质，因此研究代数簇的双有理不变量有着悠久的传统，例如Albanese簇（和/或其Tate模），Unramified Brauer群等。 另一方面，Anabelian几何从伽罗瓦理论的角度研究算术和/或几何。此外，双有理阿阿贝尔几何近年来又有了很大的复兴，主要是在几乎阿贝尔的背景下，它使用了可以上同调研究的基本群的幂零截断。本研究将应用阿那贝耳几何的工具，并发展新的工具，透过上同调的透镜，来研究函数场及其几何模型的算术/几何。除了伽罗瓦上同调，一个中心球员在拟议的研究是“通用上同调”的功能领域。这取决于上同调理论的选择，但是各种比较同构在一般上同调的实现之间诱导出相应的比较同构。虽然类属上同调在许多方面类似于伽罗瓦上同调，但它经常继承额外的结构，例如混合霍奇结构或绝对伽罗瓦群的作用，这使这些群具有更丰富和更精细的结构。现在已知（参见[40]），高维函数场的同构类型完全由其具有有理系数的泛上同调环确定，赋予1次正则混合Hodge结构。因此，一旦我们给H^1加上一些额外的基元数据，所有的双有理不变量都被编码在类属上同调环中，尽管是以一种非常间接的方式。此外，通有上同调群通常配备有给定函数域的\ell-adic伽罗瓦上同调的规范比较态射，它在\ell-adic完成后成为同构。在这方面，一般上同调可以被视为之间的桥梁伽罗瓦理论的背景下，这是服从anabelian技术，和motivic上下文，其中（双有理）不变量编码的几何意义的方式。拟议的研究的总体目标是利用这座桥梁在一个双向的方式：阿纳贝尔技术将被用来调查motivic对象，而motivic结构将被用来获得新的见解阿纳贝尔几何。

Algebraic and arithmetic geometry study solutions to polynomial and/or diophantine equations, while Galois theory studies the symmetries of such equations. By considering these arithmetic/geometric objects up-to birational equivalence, one can often obtain new information about the relationships between them. In fact, many important questions in arithmetic/algebraic geometry are of birational nature, and so there is a longstanding tradition of studying birational invariants of algebraic varieties, such as the Albanese variety (and/or its Tate module), the unramified Brauer group, etc. Several important open problems, such as the Tate conjecture for divisors, are also known to be birationally invariant. On the other hand, anabelian geometry studies arithmetic and/or geometry from the point of view of Galois theory. Moreover, birational anabelian geometry has seen a major resurgence in recent years, primarily in the almost-abelian context, which uses nilpotent truncations of fundamental groups that can be studied cohomologically. The proposed research will apply tools from anabelian geometry, and develop new ones, in studying the arithmetic/geometry of function fields and their geometric models, through the lens of cohomology. In addition to Galois cohomology, a central player in the proposed research is the "generic cohomology" of function fields. This depends on a choice of a cohomology theory, but the various comparison isomorphisms induce corresponding comparison isomorphisms between the realizations of generic cohomology. Although generic cohomology resembles Galois cohomology in many ways, it often inherits additional structure, such as a mixed Hodge structure or an action by an absolute Galois group, which gives these groups a richer and more refined structure. It is now known (see [40]) that the isomorphy type of a higher dimensional function field is completely determined by its generic cohomology ring, with rational coefficients, endowed with the canonical mixed Hodge structure in degree 1. Thus, all birational invariants are encoded in the generic cohomology ring, once one attaches some additional motivic data to H^1, albeit in a highly indirect way. Moreover, the generic cohomology groups often come equipped with a canonical comparison morphism to the \ell-adic Galois cohomology of the given function field, which becomes an isomorphism after \ell-adic completion. In this regard, generic cohomology can be seen as a bridge between the Galois-theoretical context, which is amenable to anabelian techniques, and the motivic context, where (birational) invariants are encoded in geometrically meaningful ways. The general goal of the proposed research is to utilize this bridge in a bidirectional way: anabelian techniques will be used to investigate motivic objects, while motivic structures will be used to gain new insight in anabelian geometry.